About the book
A manifold can be defined as a topological space in which the global properties may vary widely and have a complicated structure, but near each point it resembles a Euclidean space. The concept of manifold is central to modern physical mathematics as it provides tools and techniques for the analysis of complex physical systems using simple topological properties of Euclidean spaces. From Riemann's acceptance that a Euclidean space structure was suitable to study the physical knowledge of his time, until the most recent efforts to produce highly accurate higher dimensional manifolds, there has been a tremendous cross-fertilization of theory, ideas and applied numerical algorithms on manifolds based on a vivid interplay between differential geometry, topology, analysis, algebra and physics.
This book intends to provide the reader with a comprehensive overview of the current state-of-the-art in this fascinating and critically important field of mathematics, presenting some of the most important developments of the last years alongside its applications in areas such as computer-graphics, robotics, augmented-reality, machine learning, quantum mechanics and general relativity, to name only a few.